let B0 be Subset of (TOP-REAL 2); :: thesis: for K0 being Subset of ((TOP-REAL 2) | B0)
for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st f = Sq_Circ | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1 ) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1 ) ) ) & p <> 0. (TOP-REAL 2) ) } holds
( f is continuous & K0 is closed )
let K0 be Subset of ((TOP-REAL 2) | B0); :: thesis: for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st f = Sq_Circ | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1 ) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1 ) ) ) & p <> 0. (TOP-REAL 2) ) } holds
( f is continuous & K0 is closed )
let f be Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0); :: thesis: ( f = Sq_Circ | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1 ) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1 ) ) ) & p <> 0. (TOP-REAL 2) ) } implies ( f is continuous & K0 is closed ) )
assume A1: ( f = Sq_Circ | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1 ) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1 ) ) ) & p <> 0. (TOP-REAL 2) ) } ) ; :: thesis: ( f is continuous & K0 is closed )
the carrier of ((TOP-REAL 2) | B0) = B0 by PRE_TOPC:29;
then reconsider K1 = K0 as Subset of (TOP-REAL 2) by XBOOLE_1:1;
defpred S₁[ Point of (TOP-REAL 2)] means ( ( $1 `2 <= $1 `1 & - ($1 `1 ) <= $1 `2 ) or ( $1 `2 >= $1 `1 & $1 `2 <= - ($1 `1 ) ) );
{ p where p is Point of (TOP-REAL 2) : ( S₁[p] & p <> 0. (TOP-REAL 2) ) } c= NonZero (TOP-REAL 2) from JGRAPH_3:sch 1();
then A2: ((TOP-REAL 2) | B0) | K0 = (TOP-REAL 2) | K1 by A1, PRE_TOPC:28;
defpred S₂[ Point of (TOP-REAL 2)] means ( ( $1 `2 <= $1 `1 & - ($1 `1 ) <= $1 `2 ) or ( $1 `2 >= $1 `1 & $1 `2 <= - ($1 `1 ) ) );
reconsider K1 = { p7 where p7 is Point of (TOP-REAL 2) : S₂[p7] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch 1();
reconsider K2 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= p7 `1 } as closed Subset of (TOP-REAL 2) by JGRAPH_2:56;
reconsider K3 = { p7 where p7 is Point of (TOP-REAL 2) : - (p7 `1 ) <= p7 `2 } as closed Subset of (TOP-REAL 2) by JGRAPH_2:57;
reconsider K4 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `1 <= p7 `2 } as closed Subset of (TOP-REAL 2) by JGRAPH_2:56;
reconsider K5 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= - (p7 `1 ) } as closed Subset of (TOP-REAL 2) by JGRAPH_2:57;
A3: K2 /\ K3 is closed by TOPS_1:35;
A4: K4 /\ K5 is closed by TOPS_1:35;
A5: (K2 /\ K3) \/ (K4 /\ K5) c= K1

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in (K2 /\ K3) \/ (K4 /\ K5) or x in K1 )
assume A6: x in (K2 /\ K3) \/ (K4 /\ K5) ; :: thesis: x in K1

per cases ( x in K2 /\ K3 or x in K4 /\ K5 ) by A6, XBOOLE_0:def 3;

suppose x in K2 /\ K3 ; :: thesis: x in K1

then A7: ( x in K2 & x in K3 ) by XBOOLE_0:def 4;
then consider p7 being Point of (TOP-REAL 2) such that
A8: ( p7 = x & p7 `2 <= p7 `1 ) ;
consider p8 being Point of (TOP-REAL 2) such that
A9: ( p8 = x & - (p8 `1 ) <= p8 `2 ) by A7;
thus x in K1 by A8, A9; :: thesis: verum

end;

suppose x in K4 /\ K5 ; :: thesis: x in K1

then A10: ( x in K4 & x in K5 ) by XBOOLE_0:def 4;
then consider p7 being Point of (TOP-REAL 2) such that
A11: ( p7 = x & p7 `2 >= p7 `1 ) ;
consider p8 being Point of (TOP-REAL 2) such that
A12: ( p8 = x & p8 `2 <= - (p8 `1 ) ) by A10;
thus x in K1 by A11, A12; :: thesis: verum

end;

end;

end;

K1 c= (K2 /\ K3) \/ (K4 /\ K5)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in K1 or x in (K2 /\ K3) \/ (K4 /\ K5) )
assume x in K1 ; :: thesis: x in (K2 /\ K3) \/ (K4 /\ K5)
then consider p being Point of (TOP-REAL 2) such that
A13: ( p = x & ( ( p `2 <= p `1 & - (p `1 ) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1 ) ) ) ) ;
( ( x in K2 & x in K3 ) or ( x in K4 & x in K5 ) ) by A13;
then ( x in K2 /\ K3 or x in K4 /\ K5 ) by XBOOLE_0:def 4;
hence x in (K2 /\ K3) \/ (K4 /\ K5) by XBOOLE_0:def 3; :: thesis: verum

end;

then K1 = (K2 /\ K3) \/ (K4 /\ K5) by A5, XBOOLE_0:def 10;
then A14: K1 is closed by A3, A4, TOPS_1:36;
defpred S₃[ Point of (TOP-REAL 2)] means ( ( $1 `2 <= $1 `1 & - ($1 `1 ) <= $1 `2 ) or ( $1 `2 >= $1 `1 & $1 `2 <= - ($1 `1 ) ) );
{ p where p is Point of (TOP-REAL 2) : ( S₃[p] & p <> 0. (TOP-REAL 2) ) } = { p7 where p7 is Point of (TOP-REAL 2) : S₃[p7] } /\ (NonZero (TOP-REAL 2)) from JGRAPH_3:sch 2();
then K0 = K1 /\ ([#] ((TOP-REAL 2) | B0)) by A1, PRE_TOPC:def 10;
hence ( f is continuous & K0 is closed ) by A1, A2, A14, Th25, PRE_TOPC:43; :: thesis: verum